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Associated Prime Ideals in Non-Noetherian Rings Can. J. Math., Vol. XXXVI, No. 2, 1984, pp. 344-360 ASSOCIATED PRIME IDEALS IN NON-NOETHERIAN RINGS JUANA IROZ AND DAVID E. RUSH The theory of associated prime ideals is one of the most basic notions in the study of modules over commutative Noetherian rings. For modules over non-Noetherian rings however, the classical associated primes are not so useful and in fact do not exist for some modules M. In [4] [22] a prime ideal P of a ring R is said to be attached to an JR-module M if for each finite subset / of P there exists m e M such that / Q ann# (m) Q P. In [4] the attached primes were compared to the associated primes and the results of [4], [22], [23], [24] show that the attached primes are a useful alternative in non-Noetherian rings to associated primes. Several other methods of associating a set sé(M) of prime ideals to a module M over a non-Noetherian ring have proven very useful in the past. The most common of these is the set Assy(M) of weak Bourbaki primes of M [2, pp. 289-290]. Another method, which was used by Krull in 1929 [8] and later studied by Banaschewski [1] and Kuntz [9], is the following. Call a prime ideal Pa Krull prime of M if for each a e P there exists an m e M such that a e ann# (m) Q P. In his study of pseudo-Noetherian rings [14] [15], K. McDowell was led in [16, pp. 36-37] to consider the set of attached primes of M, which he called the strong Krull primes of M due to their relationship to the Krull primes. We prefer McDowell's terminology because of this connection with the Krull primes and also because the "attached prime" terminology has previously been used to mean something entirely different. (See for example the papers [12, 13, 25, 27, 29, 30, 31].) McDowell developed many of the basic properties of strong Krull primes in [16] and in [17] where he compared the strong Krull primes to other well-known types of associated primes that have been used in non-Noetherian rings. After defining and summarizing in Section 1 some of the relationships among seven different notions of associated primes which appear in the literature, we show, in Section 2, that unlike the Nagata primes and weak Bourbaki primes, the Krull primes and strong Krull primes behave well with respect to flat ring extensions. Some further results on the behavior of these primes in polynomial rings are also given. After some remarks in Section 3 on when the strong Krull primes and weak Bourbaki primes are the same, we give in Section 4 a consequence of this behavior for Received November 25, 1982 and in revised form August 5, 1983. 344 Downloaded from https://www.cambridge.org/core. 01 Oct 2021 at 18:55:49, subject to the Cambridge Core terms of use. ASSOCIATED PRIME IDEALS 345 seminormality in abelian group rings. In Section 5 we show that the main result of [23] can be strengthened by replacing the strong Krull primes by weak Bourbaki primes. In the final section we give a brief discussion of the notion of "associated prime" in general. The authors were fortunate to have had access to the unpublished manuscript [17] and gratefully acknowledge the debt that the present work owes to it. 1. Definitions and notation. We begin by giving the definitions of seven different notions of associated prime ideals that appear in the literature. For this we will need some notation. All rings will be commutative with identity. If M is an ^-module we let ZR(M) = {r e R\rm = 0 for some m e M, m ¥= 0}. If S is a multiplicative subset of R, M (S) denotes {m <E M\sm = 0 for some s e S). If S = R — P where P is a prime ideal of R we write M(P) instead of M(S). UN ç M, ami;? (N) denotes the annihilator of N(= {x <= R\xN = 0} ). We will write ann# (m) instead of ann# ( {m} ) if m e M. Definition. Let P be a prime ideal of R, and M an i^-module. (a) If P = ann# (m) for some m e M we call P a Bourbaki prime of M. We let Ass(M) denote the set of Bourbaki primes of M. See for instance [2, Chapter IV] for a discussion of these primes. (b) If P <E Ass(M/M(P) ) then P is called a Noether prime of M. The set of these primes will be denoted Ne(Af). They are discussed in [10]. (c) P is called a Zariski-Samuel prime of M if ann# (m) is P-primary for some m e M. These primes will be denoted Z — S{M) and are discussed in [33]. (d) If P is minimal over ann# (m) for some m e M, then P is called a weak Bourbaki prime of M. We will denote the set of weak Bourbaki primes of M by Ass^(M). Many of the basic properties of these primes are found in [2, Chapter IV, Section 1, exercises 17-19] and [11]. (e) P is called a strong Krull prime of M if for every finitely generated ideal / contained in P there exists an m G M such that / Q ann (m) Q P. We will let sK(M) denote the set of strong Krull primes of M. They are discussed in [17] and also in [4] [22] [23] [24] where they are called attached primes. (f) P is called a Krull prime of M if for each/? G P there exists m G M such that p e ann (m) Q P. We will denote the set of these primes by K(M). See [1], [9] for the basic properties of Krull primes. (g) If there exists a multiplicative subset S of R such that S~lP is ] maximal in ZS\R(S~ Af), then P is called a Nagataprime of M. Many of Downloaded from https://www.cambridge.org/core. 01 Oct 2021 at 18:55:49, subject to the Cambridge Core terms of use. 346 J. IROZ AND D. E. RUSH the basic properties of these primes can be found in [20] [33]. We will denote the set of Nagata primes of M by N(M). For an i^-module M we have Ass(M) ç Z-S(M) ç Ass/(M) Q sK(M) Ç K(M) Q N(M) and Ass(M) Q Ne(M) ç Assy(M). Further, all of these containments can be proper [33] [9] [10] [17], and there are no inclusions relating Ne(M) and Z — S(M). The equality Z-S(M) = N(M) holds if the zero submodule of M has a primary decomposition [20], and if R is Noetherian then Ass(M) = N(M). Also, it can happen that Ass(M), Ne(M), and Z — S(M) are empty for M ^ 0, whereas clearly Ass/(M) = 0 ** M = 0. In [9] the Krull primes of M were studied only for cyclic modules. However, most of these results extend easily to arbitrary i?-modules M. For example, if M is an i?-module and P is a prime ideal, the following statements are equivalent (see [9, Proposition 1] ): (a) P G K(M\ (b)P ç ZR{M/M(P)\ (c)P = ZR(M/M(P)). Also, if S is a multiplicative subset of R it follows that l l Ks-iR(S~ M) = {S~ P\P e KR(M), P n S = <t>). the corresponding result also holds for Assy(M) [2] and for sK(M) [4] [17]. 2. Associated primes and ring extensions. In this section <f>:A —> B will be a ring homomorphism and fl<f>:Spec(5) -> Spec(^) will denote the induced map. If M is a i?-module the A -module obtained from M by restriction of scalars will be denoted ^M. If M is an A -module, then M ®^ B is a J9-module and when we consider it as an A -module we will write $(M ®A B). In [11] Lazard showed that if M is a ^-module then Ass/^M) ç ^(AsS/(M) ) and that equality holds if <f> is flat. Thus if M is an A -module and <f> is flat then Downloaded from https://www.cambridge.org/core. 01 Oct 2021 at 18:55:49, subject to the Cambridge Core terms of use. ASSOCIATED PRIME IDEALS 347 a Ass,(*(M 0^ B) = <j>(Assf(M ®A B) ). Heinzer and Ohm [5, Example 4.4] gave an example of a flat yl-algebra $:A -> B and an A -module M having a prime fl P G #Ass/(M ®A B) ) with P £ Assy(M), i.e., weak Bourbaki primes of M ®A B do not necessarily contract to weak Bourbaki primes of M. More recently Heitmann has given an example showing that Nagata primes of M ®A B do not always contract to Nagata primes of M [6] (again where (j>:A —> B is flat). To date the only positive result obtained without further restrictions on <J> is [5, Proposition 4.5] which states that if (j>:A —> B is flat, M is a a cyclic ^-module, and P G ASS/(M ®A B), then <f>(P) G N(M). In this section we strengthen this by showing that if <J> is flat and M an A -module, then strong Krull primes of M ®A B contract to strong Krull primes of M and Krull primes of M ®A B contract to Krull primes of M.
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